| L(s) = 1 | + (−0.628 + 1.26i)2-s + (−1.28 + 0.898i)3-s + (−1.21 − 1.59i)4-s + (3.36 − 0.294i)5-s + (−0.331 − 2.19i)6-s + (−3.47 + 2.00i)7-s + (2.77 − 0.532i)8-s + (−0.186 + 0.512i)9-s + (−1.74 + 4.45i)10-s + (0.281 + 1.04i)11-s + (2.98 + 0.956i)12-s + (−1.08 + 1.54i)13-s + (−0.357 − 5.65i)14-s + (−4.05 + 3.40i)15-s + (−1.07 + 3.85i)16-s + (−2.40 + 0.876i)17-s + ⋯ |
| L(s) = 1 | + (−0.444 + 0.895i)2-s + (−0.740 + 0.518i)3-s + (−0.605 − 0.796i)4-s + (1.50 − 0.131i)5-s + (−0.135 − 0.894i)6-s + (−1.31 + 0.757i)7-s + (0.982 − 0.188i)8-s + (−0.0622 + 0.170i)9-s + (−0.551 + 1.40i)10-s + (0.0847 + 0.316i)11-s + (0.861 + 0.275i)12-s + (−0.300 + 0.429i)13-s + (−0.0955 − 1.51i)14-s + (−1.04 + 0.879i)15-s + (−0.267 + 0.963i)16-s + (−0.584 + 0.212i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00935803 + 0.613042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00935803 + 0.613042i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.628 - 1.26i)T \) |
| 19 | \( 1 + (4.12 - 1.42i)T \) |
| good | 3 | \( 1 + (1.28 - 0.898i)T + (1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (-3.36 + 0.294i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (3.47 - 2.00i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.281 - 1.04i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (1.08 - 1.54i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (2.40 - 0.876i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.77 - 2.11i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.812 + 1.74i)T + (-18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (1.98 + 3.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.54 - 8.54i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.00 + 0.353i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-11.2 + 0.981i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-9.49 - 3.45i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (0.825 - 9.43i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (2.74 - 5.89i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-6.70 - 0.586i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-4.72 - 10.1i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (0.517 - 0.616i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.17 - 0.206i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.49 + 14.1i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.889 - 3.31i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-11.6 + 2.05i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.56 + 2.02i)T + (74.3 - 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.31069562451675218374629700276, −10.80134355837119110859315295604, −10.05177358025698606716665167219, −9.421405098277006102650103320102, −8.725565313336623291044693858971, −7.01711519478715841585967926591, −6.01305401390407524980300659408, −5.71901992422019672025754729393, −4.48928526818767177944067308526, −2.19359050649395181773853806817,
0.54320382607553356824876322443, 2.25881512028228930371691349264, 3.59061490538047983199124167823, 5.32460888077769720860807319612, 6.46455904234702103323123520109, 7.08854386184198726173405877541, 8.895580302367802281565487452069, 9.508963244268400035469443769800, 10.51834500191495289250053376816, 10.91427891161109525301347881062
